Hi, your parent would not be subject to the annual visa availability because they are immediate relatives of the US citizen. The whole process will take about 6-9 months.
Just FYI, by whole process we mean filing the I-130 with USCIS, having the immigrant visa interview at the embassy, and then entering the U.S. and receiving the green card from USCIS.
I agree with my colleagues with the caveat that if the parent of the US citizen has been previously deported or denied entry, there may be additional issues you need to deal with.
Hire an immigration attorney to assist you.